Cobweb Notes
John Mason 2011
Purpose
To promote appreciation:
that graphs are there to be worked on and with, not simply pictures as the end product of a task;
of the use of coordinates to make sense of graphs;
of the variation in shape that cubic and quartic graphs can take;
Applets
There are currently Versions V22, V23, V24, V34 V34A. The numbers indicate the degrees of the
polynomials involved. V34A answers a question prompted by V34 which is nevertheless available with
the others.
Buttons
The grey line other than the axes is the line y=x. Sometimes you need to click on the grey background
before clicking on a button.
The polynomial names display the associated curve with their specifying points. These can be adjusted
to see how the curve can be changed.
Pts hides the points to clear the screen
Restore redraws the original curves by putting the specifying points back where they started.
Path1 and Path2 show a construction which can be animated by the play-pause buttons bottom left, or
simply dragged by the user using the red point marked x.
Locus1 and Locus2 show the loci of the while point in the path constructions.
Alignment displays horizontal and vertical lines for emphasising alignments between extreme values of
the curves.
Images displays the x-value as a y-value for locating images of one function as input to a second
function.
Translate translates all the points specifying the curves by the black arrow, so the graphs can be shifted
around the screen. Note that translating is cumulative; to reset the functions use Restore.
Scale enables the points to be called relative to the origin: click-hold the button and type in a scale factor.
Note that scaling is cumulative; to reset the functions use Restore.
Sometime s there is a residual white-dot where the last selected point was: clicking anywhere away will
remove it.
Tasks
What is the construction indicated by animating the red ‘x’ point? Describe the construction.
What function is traced out by the white end-point of the construction?
What degree polynomial would you expect for the compound or composite function?
What alignments appear to occur between extreme values of the component and the compound or composite
functions? Are these particular or are these general?
What alignments appear to occur between where extreme values are taken? Are these particular or are these
general?
What relationships between the two initial polynomials guarantees that both composites will display their
appropriate degree?
Comments
Comments and suggestions to j.h.mason@open.ac.uk are most welcome.
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